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Journal of Mathematical Physics

On the Weyl algebras for systems with semibounded and bounded configuration space

L. Bracci, Luigi E. Picasso

Published Nov 1, 2006• 5 citations
Publisher

Abstract

We define the Weyl algebra suitable to represent the quantization postulate for one-dimensional systems whose configuration space is semibounded. It consists of a group {V(β)=exp(−iβq),β∊R} of unitary operators and a semigroup {U(α),α⩾0} of nonunitary isometries. We show that the spectrum σ(q) is a half-line [x0,∞), with an arbitrary x0>−∞, and that the irreducible representations of the Weyl algebra with the same x0 are equivalent. We also consider the case when the semigroup of translations is substituted with a semigroup of partial isometries of index 1 (particle confined to a segment of unit length). The uniqueness of the irreducible representations of the related Weyl algebra is proved also for this case by exploiting the result for the half-line.

DOI
10.1063/1.2363256

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We define the Weyl algebra suitable to represent the quantization postulate for one-dimensional systems whose configuration space is semibounded. It consists of a group {V(β)=exp(−iβq),β∊R} of unitary operators and a semigroup {U(α),α⩾0} of nonunitary isometries. We show that the spectrum σ(q) is a half-line [x0,∞), with an arbitrary x0>−∞, and that the irreducible representations of the Weyl algebra with the same x0 are equivalent. We also consider the case when the semigroup of translations is substituted with a semigroup of partial isometries of index 1 (particle confined to a segment of unit length). The uniqueness of the irreducible representations of the related Weyl algebra is proved also for this case by exploiting the result for the half-line.
The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.
summary:In the paper the origins of the intrinsic unitary symmetry encountered in the study of bosonic systems with finite degrees of freedom and its relations with the Weyl algebra (1979, Jacobson) generated by the quantum canonical commutation relations are presented. An analytical representation of the Weyl algebra formulated in terms of partial differential operators with polynomial coefficients is studied in detail. As a basic example, the symmetry properties of the ddd-dimensional quantum harmonic oscillator are considered. In this case the usual set of the creation and annihilation operators is used as basis of the Weyl algebra under consideration. From this model are deduced the underlying symplectic structure and some of its consequences. In particular, the transitions from the initial $\text{Sp}(d,\bbfC)$-invariant model to the $\text{GL}(d,\bbfC)$- and, at last, U(d)\text{U}(d)U(d)-invariant models are performed.
Starting from a (possibly infinite dimensional) pre-symplectic space (E, σ) , we study a class of modified Weyl quantizations. For each value of the real Planck parameter ℏ , we have a C*-Weyl algebra \mathcal W(E,ℏ_σ) , which altogether constitute a continuous field of C*-algebras, as discussed in previous works. For ℏ = 0 , we construct a Fréchet–Poisson algebra, densely contained in \mathcal W(E, 0) , as the classical observables to be quantized. The quantized Weyl elements are decorated by so-called quantization factors, indicating the kind of normal ordering in specific cases. Under some assumptions on the quantization factors, the quantization map may be extended to the Fréchet–Poisson algebra. It is demonstrated to constitute a strict and continuous deformation quantization, equivalent to the Weyl quantization, in the sense of Rieffel and Landsman. Realizing the C*-algebraic quantization maps in regular and faithful Hilbert space representations leads to quantizations of the unbounded field expressions.
It is possible that a complete physical theory can be written entirely in terms of operators such as current densities, rather than in terms of field operators. The current densities in these models correspond in general to distributions of unbounded operators. Such a theory is reviewed for the case of nonrelativistic quantum mechanics. It is found that one can exponentiate the current algebra to obtain a group, which may then be represented by unitary (hence bounded) operators in Hilbert space. This procedure is analogous to exponentiating the canonical commutation relations and obtaining the Weyl group. For nonrelativistic quantum mechanics without spin, the group is the semidirect product J∧K Schwartz space J(R3) with a group K of certain C∞ diffeomorphisms from R3 onto itself. For f ∈ J and φ ∈ K, the composition mapping (f,φ)→f∘φ defines the semidirect product law. The Gel'fand-Vilenkin formalism for ``nuclear Lie groups'' is suitable for the representation theory of such a group. Almost all of the physical information is contained in a cylindrical measure μ on J′, the dual of the nuclear space J. The Fourier transform of μ can be interpreted as an expectation functional with respect to the state of lowest energy. The nonrelativistic Fock representation (including the theory of n particles in a box) is examined in this formalism. Conditions on μ are systematically developed which suffice to recover all of the infinitesimal generators on a common, dense, invariant domain in the Hilbert space. For the Weyl group slightly weaker conditions than for the semidirect product case would suffice. Gaussian measures in J′, as well as the measures defining n-particle representations of J ∧K, satisfy these conditions.
In this paper we describe and apply integral quantization, a procedure based on operator-valued measures and the resolution of the identity.We explore its covariance properties in the context of group representation theory.Three applications based on group representations are carried out.The first one concerns the covariant integral quantization based on the spin one-half irreducible representation of SU(2).In this case we show that the quantization of both the quadratic Hopf variables and the Euler angles reproduces the SU(2) Lie algebra.The second example revisits integral quantization based on the Weyl-Heisenberg group.By completing previous results, we show the universality of the canonical commutation relation in such quantizations.In the last example we revisit and enrich the integral quantization based on the affine group of the real line and we give a short account of a relevant application.